For Clark, it would mean an end to the sort of private life he wanted for himself. It would mean an end of him not having to be perceived by everyone around him as a god-like hero first and a person second. Sure, he can still go to movies or chill out in a park, but he'll never be able to relax or be free of the ancillary aspects of being a savior superhero. People will be drawn to him to be close to a celebrity, to ask for help, or to spew invective. His future as a journalist or any sort of objective public voice will be in question if not totally off the table.
I didn't say a single thing about the necessity of seeing Clark change in a phonebooth. That sort of trivial background noise of the secret identity is not what having a private life as Clark Kent means to Superman or to Superman stories. It's much more than that, and I really think that you know that already. The duality at the core of Superman is between a larger than life alien superhero and just an ordinary guy trying to do good the way ordinary people do. As soon as those lines get blurred, then the core duality has shifted to something entirely new. It will be less extraordinary/ordinary and more public/private. There will be very little "ordinary" left, but just varying shades of extraordinary. It'll be stories about Superman being "on the job" or "off the job" rather than superhero time or human time.Is duality at the core of Superman? Yes. Is the only way to explore that by knowing that he changes in a telephone booth? I'm not convinced.